On the semantics and pragmatics of epistemic …ssmoss/Moss - Semantics and...Forthcoming in Semantics and Pragmatics. Penultimate version. On the semantics and pragmatics of epistemic

Forthcoming in Semantics and Pragmatics. Penultimate version.

On the semantics and pragmatics of epistemic vocabulary

Sarah Mossssmoss@umich.edu

There has been much recent debate over the correct semantics for epistemic vo-cabulary, i.e. expressions like the sentential operators in sentences such as:

(1) John might be in his office.

(2) John must be in his office.

(3) John is probably in his office.

(4) If John is in the building, he is in his office.

This paper explores a rich source of data for theories of this vocabulary. The debateover the viability of standard truth conditional theories has called attention to thedistinctive behavior of epistemic vocabulary in eavesdropping judgments, indicativesuppositions, and statements of disagreement and retraction. But extant accountsare not sufficiently sensitive to distinctive features of the way in which epistemicvocabulary interacts with other epistemic vocabulary. If we start by studying thebehavior of simple nested epistemic modals, we may naturally build a theory thatexplains the more complicated behavior of epistemic modals under disjunction andover indicative conditionals, and even the puzzling effects of embedding epistemicvocabulary in classically valid arguments. In §1, I make unifying observations aboutthe suggestive behavior of epistemic vocabulary in each of these contexts, extractingseveral desiderata for semantic and pragmatic theories.

1. Thanks to Fabrizio Cariani, Josh Dever, Cian Dorr, John Hawthorne, Eric Swanson, Brian Weatherson,and an anonymous referee for feedback on drafts of this paper. Thanks also to the University of ChicagoLinguistics and Philosophy Workshop, the University of Michigan Linguistics and Philosophy Work-shop, Ohio State University, and the 24th Semantics and Linguistics Theory Conference (SALT 24) forhelpful discussion.

In §2–3, I develop a semantics for epistemic vocabulary. This semantics constitutesa rather dramatic alternative to standard truth conditional theories, as it assigns setsof probability measures rather than sets of worlds as semantic values. I aim to demon-strate that what my theory lacks in conservatism is made up for by its strength. In§4, I argue that combined with a novel pragmatics, my semantic theory can accountfor the distinctive linguistic behavior observed in §1. The theory I defend therebyaddresses several challenges raised in recent literature. For instance, the theory an-swers concerns about epistemic modals under disjunction raised in Schroeder 2012.The theory explains why epistemic vocabulary produces invalid instances of classi-cally valid arguments, shedding light on important puzzles raised for constructivedilemma arguments in Kolodny & MacFarlane 2010 and modus tollens argumentsin Yalcin 2012b.

1. Data for a theory of epistemic vocabulary

A careful examination of the behavior of epistemic modals yields several desideratafor a theory of epistemic vocabulary. A few of these desiderata have been discussedelsewhere, usually as puzzles concerning epistemic modals. A number of the desider-ata make trouble for extant semantic theories. The literature on epistemic modals isso vast that it would be impractical to argue against every alternative to my preferredtheory here. For considerations of space, I set aside the possibility of resuscitatingthe standard truth conditional semantics for epistemic vocabulary, since persuasivearguments against that semantics have been discussed at length elsewhere.2 I pointout potential challenges for other prominent theories in passing, but the main focusof this paper is the exposition and development of a positive case for my own theory.

1.1. Nested epistemic vocabulary

Nested epistemic vocabulary occurs in many forms in ordinary conversation. Forexample, suppose Alice and Bob are both candidates for certain job positions. Wemay naturally talk about Bob using epistemic adjectives under epistemic operators:

(5) Alice is a likely hire, and Bob might be a likely hire.

(6) Alice is a possible hire, and Bob is probably also a possible hire.

2. For instance, see the implications of triviality results discussed in Edgington 1995, the discussion ofthe subject matter of indicative conditionals in Bennett 2003, the “speaker inclusion constraint” inEgan et al. 2005 and Weatherson 2008, the case of the missing car keys in Swanson 2006 and vonFintel & Gillies 2011, the eavesdropping cases in Egan 2007, the discussion of embedding behaviorin Yalcin 2007, the discussion of inference patterns in Yalcin 2010, the discussion of assertability anddisagreement in Yalcin 2011, and the discussion of retraction and disputes in MacFarlane 2011.

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And we could further spell out the above observations as follows:

(7) It is likely that we will hire Alice, and we might also be likely to hire Bob.

(8) We might hire Alice, and it is probably the case that we might hire Bob too.3

Both epistemic modals and epistemic comparative adjectives can occur in the scopeof indicative conditionals, and vice versa:

(9) If they did not hire Alice, they are more likely to have hired Bob than Carl.4

(10) It is more likely than not that the vase broke if he dropped it on concrete.

In addition, there are well-known examples of right-nested and left-nested indicatives:

(11) If a Republican wins the election, then if it’s not Reagan who wins it will beAnderson. (McGee 1985, 462)

(12) If the cup broke if it was dropped, it was fragile. (Gibbard 1981, 237)

And finally, there are attested uses of nested epistemic expressions occurring in shortsuccession:

(13) She could not but think [that] Wentworth was not in love with either. Theywere more in love with him; yet there it was not love. It was a little fever ofadmiration; but it might, probably must, end in love with some.5

(14) The time is now near at hand which must probably determine, whether Amer-icans are to be, Freemen, or Slaves.6

In wordy constructions such as (7) and (8) as well as condensed constructions suchas (13) and (14), we are intuitively using nested epistemic modals to say somethingdifferent from what we would use single modals to say. For example, intuitively (5)says something different about Bob than it says about Alice:

(5) Alice is a likely hire, and Bob might be a likely hire.

To take another simple example, (15) intuitively says something different about Bobfrom either (16) or (17):

(15) It is probably the case that Bob is a possible hire.

3. It cannot be taken for granted that both modals in these constructions are genuinely epistemic. However,in the next section of this paper, I give several arguments against the claim that one can always provideembedded modals with non-epistemic interpretations.

4. Hacquard & Wellwood 2012 give attested cases of epistemic vocabulary in indicative antecedents,while arguing that pragmatic considerations may limit the distribution of epistemic vocabulary in in-dicative antecedents and similar linguistic contexts.

5. Austen 1818, p.55; italics added.6. George Washington’s address to the Continental Army before the Battle of Long Island, 27 August 1776;

italics added.

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(16) It is probably the case that Bob is a hire.

(17) Bob is a possible hire.

In particular, our judgments suggest that (15) is weaker than either (16) or (17). Be-lieving (16) is intuitively sufficient reason to bet at even odds that we will hire Bob,whereas merely believing (15) is not. Evidence for the semantic difference between(15) and (17) comes from direct intuitions about what we use these sentences to talkabout. In particular, nested epistemic modals are often used when you do not yethave some settled opinion on some question. If you say that Bob is a possible hire, itsounds as if you know that we might hire Bob. By contrast, if you merely say that itis probably the case that Bob is a possible hire, it sounds as if you have not yet settledon an opinion about Bob. Either Bob is a possible hire, or he isn’t, and you are moreinclined to side with the former opinion.

Relatedly, subjects sometimes report that they can easily make sense of nestedepistemic modals by imagining that the speaker has several sources of informationabout their prejacent, and she is not sure which source she should trust. For instance,suppose we survey several equally informed experts about whether we might hireBob. If most say that we might hire Bob and just a couple of experts disagree, thenit is natural to form the opinion that it is probably the case that we might hire Bob.And analogous generalizations hold for other uses of nested epistemic modals. Tocomment on the example (14) above: if you say that some battle must probably bedecisive, it sounds as if whatever settled opinion you may eventually have about theimportance of the battle, you will settle on an opinion according to which the battleis probably decisive. It is easy to make sense of this state by imagining that you haveseveral sources of information about whether the battle will be decisive, where eachsource agrees that the battle is at least more likely than not to be decisive.

According to naïve orthodoxy, when someone utters a declarative sentence, youshould add its content to your stock of full beliefs. But as theorists have developedalternatives to full belief models of mental states, many have argued that what we sayreflects what we think according to these more intricate models. For instance, somehave claimed that epistemic modals are used to communicate partial beliefs.7 At a firstglance, it may appear that sentences containing nested epistemic modals are used tocommunicate even more intricate mental states. In particular, according to imprecisecredence models, you are associated with multiple probability measures when you areunsettled as to how likely various propositions are, exactly as you might be when youare unsure what source of information you should trust. Rothschild 2012 argues that

7. See §2 for further discussion, and see Swanson 2012 for a recent catalog of relevant literature.

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epistemic modals are used to communicate these sorts of imprecise credal states. Thetheory I develop does not model subjects as having imprecise credences. But whetheror not we adopt the sort of semantics Rothschild defends, it is important that ourtheory account for intuitive judgments that naturally lend themselves to that proposal.In other words, the above discussion highlights an important goal for any theoryof epistemic vocabulary. This is our first desideratum: our theory should explainwhy nested epistemic modals signal that different opinions about some subject are inplay. Relatedly, our theory should explain why we sometimes easily make sense ofembedded modals by imagining that a speaker bases her opinions on multiple sourcesof information.

A second desideratum for our theory of epistemic vocabulary is inspired by Yal-cin 2007. Yalcin points out that our theory of epistemic possibility modals shouldexplain why conjunctions of pairs of sentences such as (18) and (19) sound bad, andwhy such conjunctions continue to sound bad when embedded under indicative sup-position, as in (20) and (21):

Some detectives are discussing the identity of a certain masked murderer.

(18) It is not John.

(19) It might be John.

(20) #Suppose it is not John and it might be John.

(21) #If it is not John and it might be John. . .

Along the same lines, note that not only is it bad to assert (18) and (19) together,but it is difficult to imagine a single circumstance in which you could correctly uttereither of these sentences individually. If you would be correct in uttering (18) in somecircumstance, then it is difficult to imagine how you could simultaneously be just ascorrect in uttering (19).

In this last respect, (18) and (19) stand in striking contrast to a similar pair ofsentences, namely sentences that resemble (18) and (19), but where the embeddedsentence is replaced with a sentence containing epistemic vocabulary:

(22) It is not the case that it is probably John.

(23) It might be the case that it is probably John.

It is possible to imagine a single circumstance in which you could correctly utter either(22) or (23). For instance, suppose you simply cannot make up your mind about howlikely it is that the masked murderer is John. A few experts believe it is probably John,but a majority of experts believe it is probably Mary. In this case, you might correctly

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use (22), insofar as you would side with the majority of experts if forced to chooseone suspect. But you might also correctly use (23), insofar as you refuse to simplyignore the minority expert opinion. Here different frames of mind are relevant toyour imagined utterances: (22) reflects your opinion after collating the advice of yourexpert advisors, while (23) reflects the fact that you are still not sure which expertsyou should trust. And of course, neither frame of mind vindicates the assertion ofboth sentences:

(24) #It is not the case that it is probably John and it might be the case that it isprobably John.8

These judgments yield a second desideratum for our theory of epistemic vocabulary:our theory should explain why in certain circumstances, we could correctly uttereither (22) or (23), though we could not correctly utter their conjunction.

A third desideratum comes from a final observation about nested modals, namelythat the strength of the outer modal often reflects the weight of your evidence and re-silience of your opinion about the prejacent of the inner modal. For example: supposethat Liem likes wearing green shirts. His dad Eric has observed the color of his shirton 800 consecutive days. Liem was wearing green on 500 of those days. His friendMadeleine has observed the color of his shirt on 8 consecutive days. Liem was wear-ing green on 5 of those days. Suppose that Eric and Madeleine have not yet seenwhat Liem is wearing today. Both Eric and Madeleine have .625 credence that Liemis wearing green, and both might guess that Liem is probably wearing green. But itseems more appropriate for Madeleine to assert (25) or (26), whereas Eric is intuitivelylicensed in asserting (27):

(25) It might be probable that Liem is wearing green.

(26) In fact, I’m fairly confident that he is probably wearing green.

(27) Liem is definitely likely to be wearing green.

The assertability of (27) tracks two differences between Eric and Madeleine. Eric baseshis credences about Liem on more evidence. In addition, his high credence that Liemis wearing green is more resilient. Joyce 2005 argues that in a number of evidentialsituations, “weight of evidence manifests itself in the resilience of credences in theface of new data” (166). In the above situation, both evidential weight and credalresilience are manifested in the strength of the modal that embeds (28):

(28) Liem is probably wearing green.

8. A less stilted but equally infelicitous version of the sentence: ‘John isn’t a probable killer and might bea probable killer’.

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Suppose you have a relatively uninformed hunch that Liem is probably wearing green.In other words, suppose that your high credence that Liem is wearing green is notjustified by much evidence. Then you are intuitively licensed in asserting (25), butnot (27). As you acquire more and more evidence, your high credence that Liem iswearing green will become more and more resilient, and you may embed (28) understronger and stronger epistemic modals. Hence our third desideratum: our theory ofepistemic vocabulary should explain this intuitive connection between nested modals,evidential weight, and credal resilience.

All three of the above desiderata pose challenges for several extant theories ofepistemic modals. For example, consider the following standard dynamic semanticentries for epistemic possibility and necessity modals:9

c[♦φ] = {w ∈ c : c[φ] 6= ∅}

c[�φ] = c \ {w ∈ c : (c \ c[φ]) 6= ∅}

From these definitions, we can derive that c[�♦φ] = c[♦φ]. Hence according to thissemantics, any string of possibility and necessity modals is equivalent to its innermostmodal. Some dynamic semanticists explicitly embrace this result, claiming that “em-bedding an epistemic modal under another epistemic modal does not in general haveany interesting semantic effects” (Willer 2013, 12). The same result holds for a promi-nent competitor of the dynamic semantic proposal, namely the semantics defendedin Yalcin 2007. As Yalcin explains: “iterating epistemic possibility operators addsno value on this semantics. . . This may explain why iterating epistemic possibilitymodals generally does not sound right, and why, when it does, the truth-conditionsof the result typically seem equivalent to ♦φ. I will generally ignore iterated epistemicmodalities” (994). It is difficult to see how semantic proposals in this spirit could suc-cessfully explain the pervasive nature of nested modals, much less account for theirdistinctive behavior.

1.2. Against contextualist re-interpretations of nested epistemic vocabulary

The most substantive recent attempt at a more responsive semantics for nested epis-temic modals appears in Yalcin 2009, where Yalcin admits that sometimes nestedmodals do “allow for coherent interpretations not equivalent to corresponding ex-pression with the most narrow modal. The latter case is not provided for by theabove semantics. In such cases I would be inclined to appeal to tacit shifting of the

9. For canonical instances of semantic proposals along these lines, see Stalnaker 1970, Veltman 1996,Beaver 2001, von Fintel & Gillies 2008b, and Willer 2013.

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information state parameter, akin to free indirect discourse” (21). For further elabo-ration, we are directed to the following passage in Yalcin 2007: “interpretation mayinvolve a tacit shift in the information parameter. . . to the target state of informationfor the context. Aside from Gricean considerations of charitable interpretation, it isnot obvious whether general principles are involved in the interpretation of such tacitshifts” (1013). It is difficult to know exactly what is intended by these brief sugges-tions, and hence my arguments so far may be understood as an invitation to developthese suggestions into a theory that satisfies the desiderata given above.

A natural development of these suggestions might say that in any sentence wherenested modals occur, the prejacent of the outer modal receives the same boring sort ofsemantic value as any simple declarative sentence. For instance, one might assimilatesentences such as (27) with sentences about particular probability functions, such as(29) or (30):

(27) It is almost certainly the case that Liem is probably wearing green.

(29) It is almost certainly the case that the objective chance that Liem is wearinggreen is high.

(30) It is almost certainly the case that my epistemic probability that Liem iswearing green is high.

However, there are many reasons to be skeptical of this approach. Recall that recentliterature has provided a host of reasons to reject the claim that the prejacent (28) isequivalent to some simple declarative sentence like (31) or (32):

(28) Liem is probably wearing green.

(31) The objective chance that Liem is wearing green is high.

(32) My epistemic probability that Liem is wearing green is high.

The crucial dialectical point to appreciate is that analogous concerns tell against theequivalence of these same sentences when they are embedded under epistemic vocab-ulary. For example, it is suspiciously difficult to say exactly what salient probabilityfunction (27) is talking about. In the case described above, Eric can utter (27). But hecannot utter (29), because Eric knows that the objective chance that Liem is wearinggreen is either 0 or 1, and Eric is not almost certain of the latter. Madeleine cannotutter (27). But she can utter (30), because she knows that her inductive evidence con-firms the claim that Liem is wearing green. Hence neither (29) nor (30) accuratelyparaphrases (27).

Furthermore, eavesdroppers may explicitly target the prejacent of (27) and cor-rectly evaluate it relative to their epistemic situation. For instance, if I have just seen

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Liem wearing a red shirt and I overhear Eric utter (27), it would be pedantic butnevertheless acceptable for me to say:

(33) That isn’t almost certain; it’s just false. It’s not the case that Liem isprobably wearing green—he is wearing red.

A notorious dilemma for truth-conditional accounts replays itself here: if Eric wasusing ‘probably’ just to talk about his own evidential situation, then I am not licensedin saying ‘it’s false’ in judging the prejacent of (27). On the other hand, if Eric wasusing ‘probably’ to talk about some evidence that included my evidence, then he wasnot licensed in uttering (27) to begin with.10

In fact, nearly every argument against a uniform truth conditional theory of allepistemic modals yields an analogous argument against a uniform truth conditionaltheory of all embedded epistemic modals. Bennett 2003 may argue that any allegedparaphrases of (27) fail to capture its intuitive subject matter, for instance. Bennettargues that when someone utters an indicative conditional, “common sense and theRamsey test both clamour that [she] is not assuring me that her value for a certainconditional probability is high, but is assuring me of that high value. . . She aims toconvince me of that probability, not the proposition that it is her probability” (90).Yalcin 2011 adds that the reasons that I give in support of my utterance ‘it might beraining’ concern the first-order proposition that it is raining, rather than any contex-tually determined body of evidence. Both Bennett and Yalcin could complain that(27) intuitively concerns Liem, rather than any contextually determined body of evi-dence. Another challenge comes from Yalcin 2007. If embedded modals are alwaysinterpreted relative to some salient probability function, then we lack an explanationfor the infelicity of sentences such as:

(34) #Probably, it is raining and might not be raining.

(35) #It is unlikely that it is both raining and might not be raining.

(36) #It might be that it is both raining and might not be raining.

These judgments are not accommodated by expressivist, relativist, or dynamic theo-ries that resort to assigning simple semantic contents to embedded modal construc-tions.

In addition, it is worth noting that if we reinterpret the prejacent of (27) as havingstraightforward truth conditions, we are still left with the problem of interpreting(37-b) in the following dialogue:

10. This is just the first step in an involved dialectic. For further discussion of eavesdropping argu-ments against truth-conditional accounts of epistemic vocabulary, see Egan et al. 2005, Egan 2007,Hawthorne 2007, von Fintel & Gillies 2008a, Yalcin & Knobe 2010, and MacFarlane 2011.

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(37) a. David: Is Liem probably wearing green?b. Eric: Almost certainly.

Familiar arguments challenge the claim that the unembedded (37-a) has straightfor-ward truth conditions. Furthermore, it is difficult to see why Gricean considerationsshould demand that we interpret (37-a) as containing free indirect discourse or atacitly shifted information parameter. Hence it seems we must find some way of in-terpreting (37-b) without appealing to such strategies. One would expect the resultingunderstanding of (37-b) to provide some similar understanding of (27), namely an al-ternative semantics that recognizes that ‘Liem is probably wearing green’ need notexpress a possible worlds content, whether it is embedded in a question or under fur-ther epistemic vocabulary. To sum up: it is not obvious that extant semantic theoriescan explain the behavior of nested epistemic modals. A natural way of developingpotential explanations on behalf of recent expressivist, relativist, and dynamic theo-ries meets with several challenges. Hence the behavior of nested epistemic modalsshould motivate us to look for alternative semantic theories.

1.3. Epistemic vocabulary under disjunction

A fourth desideratum for our theory of epistemic vocabulary is inspired by Schroeder2012. Schroeder argues that a semantic theory should not predict that you can asserta disjunction only if you can assert one of its disjuncts, even in special cases wheredisjuncts are stipulated to be governed by wide-scope epistemic modals. Schroederpoints out several reasons why this prediction would be bad. Here is one example:

Last night Shieva calls me to express frustration with the paper that she is workingon, and tells me that if she hasn’t finished by this morning, she’s going to consulther magic 8-ball about whether to give up and follow its advice. Since I know thatmost of the answers on her magic 8-ball are positive, when I recall our conversationfrom last night, I conclude that either Shieva finished her paper by this morning,or she probably gave up. (21–2)

In this case, the speaker can correctly assert ‘Shieva finished or probably gave up’without being able to assert either disjunct. Similarly, you can correctly assert (38)about the result of throwing a fair die, without being able to assert either disjunct:

(38) It is less than four or probably even.

In this respect, disjunctions embedding epistemic vocabulary are just like ordinarydisjunctions of simple sentences. In fact, asserting a disjunction usually implicatesthat you are not in a position to assert either disjunct. There is something especiallypeculiar about disjunctions embedding epistemic vocabulary, though. Even if you

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can deny one disjunct and you cannot assert the other, you may still be able to assertthe entire disjunction. For instance: you can assert (38) even though you can denythe second disjunct by itself, and you cannot assert the first. This does not hold fordisjunctions without epistemic vocabulary. If you can deny one half of a simple dis-junction, then disjunctive syllogism ordinarily proves that the remaining disjunct isequivalent to the entire disjunction, so one is not assertable without the other. Thisbrings us to our fourth desideratum: our theory should explain this surprising differ-ence between simple disjunctions and disjunctions containing epistemic vocabulary.

A semantics for ‘or’ is missing from Yalcin 2007, 2011, 2012b, and related pa-pers. Hence the relevant challenge for Yalcin is to state a semantics that predicts thebehavior just described.11 Substantially more progress has been made on disjunctionin the dynamic semantics literature. In fact, a number of dynamic accounts of disjunc-tion satisfy our fourth desideratum. According to these accounts, natural languagedisjunction is not commutative. Roughly speaking, the second half of a disjunction isnot interpreted relative to a global context, but rather relative to a local context thathas been updated with the negation of the first disjunct. This sort of account aims togive a uniform explanation of the local interpretation of ‘probably’ in (38) and localsatisfaction of licensing conditions for pronouns in disjunctions such as the followingfamous example from Roberts 1989:

(39) Either there is no bathroom in this house, or it is in a funny place.

Just as the licensing conditions for ‘it’ in (39) are satisfied in a local context where thefirst disjunct is false, values of contextual parameters in the second disjunct of (38) areprovided by a local context where the first disjunct is false. This explains why youmay assert (38) even when you can deny its second disjunct uttered in isolation. Thedisjunction is felicitous because its second disjunct is acceptable in all contexts wherethe negation of the first disjunct is given.

This dynamic account predicts that natural language disjunction is not commu-tative, and fans of this account often claim this predicted failure of commutativity asa benefit. For instance, they claim that a semantics for natural language disjunctionshould entail that (40) sounds bad even though (38) sounds fine:

(38) It is less than four or probably even.

(40) It is probably even or less than four.

11. Schroeder extrapolates a semantics for ‘or’ from Yalcin 2007 and criticizes that semantics for validating‘or’ exportation.

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However, it is not clear that we should want our semantics to predict this differencebetween (38) and (40).12 For instance, there are a number of contexts in which (40)seems just as good as (38), namely contexts in which certain partitions of logical spaceare salient. Consider the following case:

Alice just rolled a fair die and hid it under a cup in front of me. I see a blue cupand a red cup. The die is under the blue cup if it landed on a four, five, or six. Thedie is under the red cup if it landed on a number less than four.

Bob offers me a pair of bets. For one dollar, he will sell me a bet that pays fivedollars if the die landed on an even number. For another dollar, he will sell me abet that pays five dollars if the die landed on a number less than four. I am veryrisk averse, and I do not always bet to maximize expected returns. But staringfirst at the blue cup and then at the red cup, I judge that I would be comfortableaccepting both bets, since, as I put it, “either it is probably even, or less than four.”

The circumstances of the above case call attention to a certain partition of logicalspace: either the die landed on a number less than four, or it landed on a highernumber. Against this background, my utterance of (40) seems perfectly correct.13

In fact, some disjunctions like (40) sound fine without heavy contextual cues. Forinstance, you can assert any of the following disjunctions, even if you can deny thefirst disjunct and cannot assert the second:

(41) It’s either unlikely he was being honest with you, or he just wanted you tothink that he was lying.

(42) The next United States president will either almost certainly attempt torepeal a lot of Barack Obama’s policies, or they will be a Democrat withmore liberal views than Obama has.

(43) John is probably playing baseball, or it has been raining all afternoon.

These disjunctions seem to mean the same thing regardless of the order in which theirdisjuncts are uttered. In fact, they might just as well be written with their disjunctsarranged in a circle, without detriment to our ability to understand or evaluate them.This yields a fifth desideratum for our theory of epistemic vocabulary: our theoryshould explain why disjunctions such as (40) sound infelicitous in some contexts andfelicitous in others. And our theory should explain why reversing disjunct order doesnot affect the interpretation of disjunctions in contexts where they sound felicitous.

12. The commutativity of disjunction is controversial even among advocates of dynamic semantic theo-ries. For instance, Schlenker 2009 and Rothschild 2011 both provide theories according to whichdisjunction is commutative; their accounts are sympathetic with my discussion of the fifth desideratum.

13. Some readers may find it difficult to evaluate the artificial speech described above, especially since thesalience of an objective chance function may introduce noise in our judgments. The essential point of thepresent discussion is that contextual cues may make certain readings of epistemic vocabulary available.See §4.5 for more natural illustrations and a more detailed defense of this point.

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This fifth desideratum should give us pause before we endorse a semantic theorythat explicitly entails that natural language disjunction is not commutative. Further-more, the above dynamic explanation for why we can assert (38) seems insufficientlygeneral, since it does not explain why we can sometimes assert (40)–(43). The dy-namic proposal outlined above says that we can sometimes assert a disjunction like(38) when its second disjunct is deniable and its first disjunct is unassertable. But(40)–(43) are all sometimes assertable even when their first disjuncts are deniable andtheir second disjuncts are unassertable. According to the dynamic explanation, (38) isfelicitous because its second disjunct is acceptable in all contexts where the negationof the first disjunct is given. But for any of (40)–(43), the second disjunct is not accept-able even in contexts where the negation of the first disjunct is given. For example,the negation of the first disjunct of (40) is already given in an ordinary context wherea fair die is rolled, but the second disjunct of (40) is not acceptable in that context:

(40) It is probably even or less than four.

To sum up: several observations raise challenges for several extant dynamic semanticaccounts of the assertability of disjunctions. In particular, differences in the assertabil-ity of (38) and (40) seem sensitive to contextual factors, such as the salience of variousalternative sets. This should motivate us to doubt theories that derive differencesin assertability from context-insensitive semantic rules. Pragmatic theories are betterdesigned to account for the distinctive behavior of disjunctions embedding epistemicvocabulary.

1.4. Epistemic vocabulary over indicatives

A sixth desideratum for a theory of epistemic vocabulary is inspired by an examplein chapter 9 of Lycan 2001, which itself builds on a related discussion of subjunctiveconditionals in Slote 1978. Consider the following case:

Jill is standing on the roof of your office building. The local fire department occa-sionally hangs a net along the roof to protect workers doing construction. The netis strong enough to safely catch anyone who falls off the building. Just a few hoursago, you happened to notice that there was no net along the roof. As a result, youdo not believe that Jill is going to jump off the roof. Jill is a thrill-seeker who mightjump into a net for fun, but she definitely does not have a death wish. And withouta net, anyone who jumped off the roof would surely fall to the ground and die.

On the one hand, since you believe that there is no net along the roof, you are intu-itively justified in asserting:

(44) Probably, if Jill jumps off the building, she will die.

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On the other hand, you are confident that Jill does not have a death wish. If you wereinformed that Jill jumped off the building, you would immediately conclude that thelocal fire department must have installed a net since you last checked the roof. Withthat information in the front of your mind, you are intuitively justified in denying (44)and asserting:

(45) Probably, if Jill jumps off the building, she will live.

To make these observations more vivid, suppose someone asks you whether there isa net along the roof of the building. They may well know that you promised the firedepartment that you wouldn’t go around telling people whether or not there was anet along the roof, but they may still persist in pestering you for information. It isintuitively fine for you to respond:

(46) I cannot answer your questions directly. But I can tell you this much: it isreally likely that if Jill jumps off this building, she will die.

On the other hand, suppose someone asks you whether you believe that Jill is suicidal.Again, they may well know that you promised Jill that you wouldn’t go around tellingpeople about her mental state, but they may persist in pestering you for information.Suppose that it is common ground that anyone suicidal would simply cut away anysafety net and jump off the building in question. It is intuitively fine for you torespond:

(47) I cannot answer your questions directly. But I can tell you this much: it isreally likely that if Jill jumps off this building, she will live.

Hence the assertability of (44) does not depend only on your opinions about Jill andthe net, which we may stipulate are the same when you utter (46) and (47). It must alsobe sensitive to some factor that varies between these contexts of utterance. As withmany other examples we have considered, you are considering different questions inthese different contexts, and which question you are considering seems relevant towhich utterances are felicitous. Suppose you are considering the question of whetherthere is a net along the roof. Then since you believe that there is probably no net,you may say that it is probably the case that if Jill jumps from the roof, she will die.Suppose you are considering the question of whether Jill is suicidal. Then since youbelieve that she is probably not suicidal, you may say that it is probably the case that ifJill jumps from the roof, she will live. The sixth desideratum: our theory of epistemicvocabulary should explain this variation in the assertability conditions of (44).

There is no obvious mechanism for explaining this variation in many extant theo-ries of epistemic vocabulary. The semantic values for ‘probably’ and ‘if’ given in Velt-

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man 1996 and Yalcin 2012b do not depend on contextually determined parameters.An advocate of these semantic proposals might attribute variation in the interpreta-tion of (44) to scope ambiguity. At the level of logical form, ‘probably’ might takescope over the entire indicative conditional in (44) or just over its consequent. But thisdoes not seem like a plausible explanation of the behavior of (44), since context notonly affects our interpretation of (44), but also our interpretation of the unembeddedindicative conditional (48):

(44) Probably, if Jill jumps off the building, she will die.

(48) If Jill jumps off the building, she will die.

The unembedded conditional is borderline assertable when we are focusing on whetherthere is a net along the roof, but definitely unassertable when we are focusing onwhether Jill is suicidal. These judgments suggest that the interpretation of the indica-tive itself depends on contextually determined parameters.

A related challenge arises when we embed sentences like (44) in indicative condi-tionals. If we are talking about whether there is a net, you can correctly assert:

(49) If it is probably the case that Jill will live if she jumps, then there is a net.

If we are talking about whether Jill is suicidal, you can correctly assert:

(50) It is probably the case that Jill will live if she jumps.

However, you can never correctly assert:

(51) There is a net.

These judgments make trouble for certain semantic theories. Several dynamic andexpressivist theories say something roughly like the following: you believe a sentencewhen your credal state accepts it. And an information state accepts a conditionalwhen the closest state that accepts its antecedent also accepts its consequent. Sinceyou believe (50), your actual credal state accepts the antecedent of (49). Hence youractual credal state is the credal state closest to yours that accepts that antecedent.Since you believe the conditional (49), we should conclude that your actual credalstate also accepts its consequent (51). But this conclusion seems clearly false.14

14. In order to keep my discussion as general as possible, I will not use this formula to construct objectionsfor particular theories. The interested reader should combine the discussion of attitude verbs in §7 ofYalcin 2007 with the semantics for ‘if’ and ‘probably’ in the appendix of Yalcin 2012b. For dynamictheories, combine the standard dynamic semantics for attitude verbs in Heim 1992 with the dynamicsemantics for ‘if’ and ‘probably’ developed in §4 of Gillies 2004, §10 of Gillies 2010, or the appendixof Yalcin 2012b, replacing “closest credal state to yours that accepts the antecedent” with “result ofupdating your credal state on the antecedent” in my discussion above.

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The complex conditional (49) gives rise to our seventh desideratum: our theoryshould explain its assertability conditions. This is not a trivial endeavor. First, ourtheory must assign semantic contents to indicatives whose antecedents embed bothgraded epistemic vocabulary and other indicatives. Second, our theory must explainhow your beliefs can support asserting (49) in some contexts and (50) in others, with-out ever supporting (51). These facts intuitively depend on the context sensitivityof (49) and (50), and relevant contextual factors intuitively include facts about whatquestions are salient when each is uttered.

1.5. Epistemic vocabulary in classically valid arguments

The seventh desideratum also directs us toward one final category of useful obser-vations. If you believe both (49) and (50), it might seem that you could apply modusponens and infer that there is a net along the roof. But you are not licensed in believ-ing that there is a net along the roof. The final three desiderata concern instances ofclassically valid argument forms that seem invalid in virtue of containing epistemicvocabulary.

Suppose Carlos has rolled a fair die without telling us how it landed. A fair diehas three low numbers and three high numbers. Suppose we are considering thefollowing argument about the number Carlos rolled:

(52) a. If it is low, it is probably odd.b. It is not probably odd.c. Hence: it is not low.

This argument seems like an instance of modus tollens. But it also seems invalid. Thefirst premise seems correct, since 2 out of 3 of the low numbers are odd. The secondpremise seems correct, since it is just as likely that an even number was rolled as anodd number. But these premises do not justify our accepting the conclusion, since wehave no idea whether a low number was rolled. Several authors have made similarobservations about apparent instances of modus tollens containing epistemic modals.15

This raises a puzzle: should we say that (52) is not an instance of modus tollens, that(52) is valid, or that some instances of modus tollens are not valid? This brings us toour eighth desideratum: our theory of epistemic vocabulary should solve this puzzle.At a minimum, our theory should come equipped with a notion of consequence thatyields a verdict about whether (52) is valid. And whether or not it is valid, ourtheory should predict the apparent invalidity of instances of modus tollens containingepistemic vocabulary.

15. For related discussion, see Carroll 1894, Veltman 1985, Cantwell 2008, and especially Yalcin 2012b.

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Here is another apparently invalid argument about the number rolled:

(53) a. If it is low, it is probably odd.b. If it is high, it is probably even.c. It is either low or high.d. Hence: either it is probably odd or probably even.

Kolodny & MacFarlane 2010 discuss similar arguments, including the following:

(54) a. Either the butler did it or the nephew did it.b. If the butler did it, the murder must have occurred in the morning.c. If the nephew did it, the murder must have occurred in the evening.d. Hence: either the murder must have occurred in the morning or it

must have occurred in the evening.

These arguments seem like instances of constructive dilemma. But they also seem in-valid. For instance, just as it seems incorrect to say that the number rolled is probablyeven, it seems incorrect to say it is probably odd. So in the absence of any special con-textual cues, it seems incorrect to say that the number rolled is either probably even orprobably odd. It is neither probably even nor probably odd, but just as likely to be oneor the other. This brings us to our ninth desideratum: our theory should say whether(53) is valid. And whether or not it is valid, our theory should predict the apparentinvalidity of instances of constructive dilemma containing epistemic vocabulary.

Similar problems arise not just for modus tollens and constructive dilemma, butalso for disjunctive syllogism:

(55) a. It is low or probably even.b. It is not probably even.c. Hence: it is not low.

And contraposition of indicative conditionals:

(56) a. If it is low, it is probably even.b. Hence: if it is not probably even, it is not low.

Furthermore, it seems entirely appropriate to give similar explanations for the appar-ent invalidity of these inferences. Kolodny & MacFarlane 2010 and Yalcin 2012b,for instance, defend semantic theories according to which each of the relevant infer-ence rules is literally invalid. In fact, Kolodny and MacFarlane go so far as to say thatmodus ponens itself is an invalid rule of inference.

Anyone rejecting classically valid inference rules bears the burden of explainingwhy we successfully use them in ordinary reasoning. The easiest way to discharge

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this burden is by proving that the rules are indeed valid when restricted to premisesof a certain form. At a minimum, setting aside complications involving adverbs ofquantification, it seems our theory should predict that arguments are valid when theycontain no epistemic vocabulary at all. This condition raises an important question,namely exactly which arguments containing epistemic vocabulary are valid.

Kolodny & MacFarlane 2010 defend inferences involving conditionals whoseconsequents do not contain any epistemic vocabulary. However, some inferences in-volving conditionals whose consequents contain epistemic vocabulary are intuitivelyvalid as well. For instance, Yalcin 2012b suggests that the following inference isvalid:

(57) a. If the marble is big, then it might be red.b. It is not the case that it might be red.c. Hence: it is not big.

In addition, some probabilistic inference rules are intuitively valid, and some of thoserules govern indicatives with consequents embedding epistemic vocabulary. In fact,we just considered inferences of this sort in §1.4. The following inference licenses mysaying (58-c) when discussing whether there is a net along the roof:

(58) a. Probably, there is no net along the roof.b. If there is no net along the roof, then if Jill jumps, she will die.c. Hence: probably, if Jill jumps, she will die.

And the following licenses my saying (59-c) when discussing whether Jill is suicidal:

(59) a. Probably, Jill is not suicidal.b. If Jill is not suicidal, then if Jill jumps, she will live.c. Hence: probably, if Jill jumps, she will live.

This brings us to our tenth and final desideratum for a theory of epistemic vocabulary.Insofar as our theory says that standard inference rules are generally invalid, it shouldexplain why substantial classes of restricted rules appear to be genuinely valid. Inparticular, our theory should explain why (57), (58), and (59) are apparently valid,even though these inferences are riddled with epistemic vocabulary.

2. A basic semantics for epistemic vocabulary

Before stating specific semantic entries, it will be helpful to outline the basic idea ofthe semantic theory itself. Recall that in a certain context, you may correctly describethe outcome of rolling a fair die by saying:

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(40) It is probably even or less than four.

The imagined context of (40) is somewhat contrived. In particular, the context iscontrived to make a certain partition of logical space especially salient. The partitionhas two elements: either the number rolled is low, or it is high. As a result, thereare also two kinds of salient credence distributions when you utter (40). First, youhave conditional credences, conditional on the partition propositions. For example,you have higher than .5 credence that the number rolled is even, conditional on itbeing high. Second, you have a credence distribution over the partition propositionsthemselves. For example, you have .5 credence that number rolled is high. In otherwords, there are various opinions you might have after learning some informationfrom the contextually salient partition. And on top of that, you have some opinionsabout the likelihood of each bit of information that you could learn.

A first pass at my semantics: the latter opinions are associated with highermodals, while the former are associated with embedded modals. For example, itwould sound fine for you to say (60) in the context mentioned above:

(60) It might well be that the number is probably even.

According to my semantics, that is roughly because you could learn some salientinformation—namely that the number rolled is high—confirming an opinion thatgives most of its credence to the number rolled being even. To take another exam-ple, suppose that you are torn between various ways of evaluating candidates for anacademic position. It is not clear how to weigh teaching experience against researchquality, for instance, and you are open to information that would decide this questionin different ways. In spite of your indecision, you might say:

(61) It must be the case that Bob is a possible hire.

According to my semantics, that is roughly because as far as your credences areconcerned, any salient information would support an opinion that gave at least somecredence to Bob being hired. Again, the embedded modal (‘possible’) is associatedwith your credences conditional on various propositions (about ways of evaluatingcandidates), while the higher modal (‘must’) is associated with your credences inthose propositions themselves.

According to a traditional account of assertion, an assertion is “something likea proposal” (cf. Stalnaker 2010, 152), namely the proposal that the content of theassertion be added to the propositions taken for granted in the conversation. In aparadigmatic case of assertion, you believe a proposition, you assert some sentencewith that proposition as its content, and as a result, I come to believe that same

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proposition. This model of assertion fits well with a certain model of our mental life,according to which full beliefs are the opinions we have and the opinions we want toshare with each other. Meanwhile, theorists have developed alternate models of ourmental life in which degreed beliefs play a central role. It is natural to wonder whetherwe can update our account of assertion to fit these more sophisticated models.

The updated account: an assertion is like a proposal, not about a proposition thatyou should believe, but rather about a property that your credences should have. Itis still true that in a paradigmatic case of assertion, you have an opinion, you assertsome sentence with that opinion as its content, and as a result, I come to have thesame opinion. But the relevant opinions are degreed. In other words, having anopinion amounts to having credences with a certain property. The content of a declar-ative sentence is a property that credences can have. Formally, contents are sets ofprobability measures. In a paradigmatic case of assertion, when you assert a sentencewith a certain content, I come to have a credence distribution that is contained in thatcontent. For instance, you may assert a sentence whose content is the set of all mea-sures that assign probability greater than .5 to the proposition that it is raining. Onhearing your assertion, I will come to have more than .5 credence that it is raining.Following Swanson 2006, we may conceive of the content of a sentence as a constrainton credences, namely the constraint that my credences generally end up satisfying onhearing your assertion of that sentence.

Sentences containing epistemic vocabulary are context sensitive. In other words,which set of measures is the content of a sentence depends on what context you areusing the sentence in. In particular, context contributes partitions of logical space tothe semantic values of such sentences. The contextually determined partitions makethe contents of sentences more interesting. A second pass at the heart of my semantics:some asserted contents are straightforward constraints on credences, such as assign-ing greater than .5 credence to some particular proposition. But asserted contentscan also constrain your credences to have more fine-grained properties. In particu-lar, they can constrain the structure of your credences with respect to propositions innon-trivial contextually determined partitions. The content of a sentence containingnested epistemic modals will be a constraint having to do with your credences inthose propositions, and also with your credences conditional on those propositions.Higher modals correspond to the former sort of constraint, while embedded modalscorrespond to the latter.

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2.1. A semantics for logical operators

In addition to formal semantic entries, it will be useful to have some shorthand forsaying what expressions mean. Let us say that your credences satisfy the constraintthat a certain proposition accepts that it is probably raining just in case it is probably rain-ing according to your credences conditional on that proposition, or in other words,just in case your conditional credences are contained in the content that it is probablyraining. If context determines a partition of logical space, we can quantify over themembers of that partition as if they were each identified with different people. Forinstance, given a contextually determined partition, let us say that your credences sat-isfy the shorthand constraint that someone accepts that it is probably raining just in casesome proposition in the partition accepts that it is probably raining. In general, letus say your credences satisfy the constraint that someone accepts a particular contentjust in case there is some proposition in the partition such that your credences giventhat proposition are contained in that particular content. Your credences satisfy theconstraint that everyone accepts a content just in case every proposition in the parti-tion is such that your credences given that proposition are contained in that content.And so on. Rather than always explicitly describing your credences conditional onpropositions in a contextually determined partition, we have a handy shorthand thatcaptures the sense in which your credences conditional on different partition elementsoften correspond to different states of opinion that you have not yourself decided be-tween. In a rough sense, one may imagine the shorthand expressions ‘someone’ and‘everyone’ as quantifying over different sides of yourself.16

Now for the semantics. In contrast with a number of extant theories, it is straight-forward to start with a semantics for all basic logical operators, including naturallanguage disjunction. For instance: your credences are contained in the content ofa disjunction just in case every proposition in the corresponding contextually deter-mined partition is such that your credences conditional on that proposition are con-tained in the content of one of the disjuncts. The semantic entries for ‘and’ and ‘not’are predictable variants. In shorthand:

‘S or T’ means that everyone accepts that S or accepts that T.

‘S and T’ means that everyone accepts that S and accepts that T.

16. In what follows, I often simplify my discussion by just talking about whether certain partition elementsaccept a certain constraint. It should be understood that strictly speaking, whether a proposition acceptsa constraint is relative to a measure, e.g. that your credences may satisfy the constraint that someoneaccepts that it is probably raining, while my credences fail to satisfy this same constraint.

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‘not S’ means that no one accepts that S.17

In more formal vocabulary:

[[ori]]c = [λS . λT . {m : ∀p ∈ gc(i), m|p ∈ S or m|p ∈ T}]

[[andi]]c = [λS . λT . {m : ∀p ∈ gc(i), m|p ∈ S and m|p ∈ T}]

[[noti]]c = [λS . {m : ∀p ∈ gc(i), m|p /∈ S}]

A number of notes about the formal vocabulary are in order. The variable p rangesover sets of worlds, and m ranges over probability measures. The measure m|p is theresult of conditionalizing the measure m on the proposition p. Let us stipulate that Sis the semantic type of sets of measures. In the above entries, the variables S and Trange over values of type S. The logical operators ‘and’ and ‘or’ have semantic valuesof type 〈S, 〈S, S〉〉, whereas ‘not’ has a semantic value of type 〈S, S〉. For example, thecontent of a disjunction is a set of measures, as is the content of each disjunct.

Exactly which set of measures is the content of a disjunction depends on whatpartition context contributes to its content. Following Heim & Kratzer 1998, we saythat every context c determines an assignment function gc that specifies the values ofall contextually determined variables. The value gc(i) is the contextually determinedpartition relevant to the semantic entry spelled out above. The shorthand expression‘everyone’ corresponds to the formal expression ‘∀p ∈ gc(i)’ which quantifies overpropositions in that partition. In what follows, I use both shorthand and formalvocabulary, as the former allows me to make my arguments intuitive, while the latterallows me to make them precise.

In slightly less formal vocabulary, the semantic value of ‘S or T’ is the set ofmeasures m satisfying the following condition: for any proposition p in the relevantcontextually determined partition, m|p is either contained in the content of the firstdisjunct or in the content of the second disjunct. For example, recall that in somecontexts where you have equal credence in each possible outcome of rolling a fair die,it sounds okay for you to say:

(40) It is probably even or less than four.

As mentioned earlier, the sort of context that is hospitable for (40) makes a certainpartition salient: either the number rolled is low, or it is high. According to your

17. I use ‘not’ as shorthand for ‘it is not the case that’ and I treat this expression as an operator that occursjust before its argument, though ultimately one should allow many other expressions of sententialnegation at surface structure. The analogous claims hold for ‘might’, ‘must’, and ‘probably’.

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credences conditional on it being low, the number is less than four. According toyour credences conditional on it being high, the number is probably even. Henceyour credences satisfy the content of (40), namely that everyone in the contextuallydetermined partition either accept that the number rolled is probably even or acceptthat it is less than four. In a nutshell: you believe (40), and that is why it sounds okayfor you to say it.

This explanation is incomplete as it stands. For starters, a complete explanationrequires identifying the content of each disjunct of (40) relative to the sort of contextin question, so that we may prove that your conditional credences are contained inthese contents. Appendix B.1 contains a complete explanation of why your credenceis in the content of (40), and §2.4 contains further commentary. Another clarificatorynote: the above semantic values are custom-made for logical operators embeddingepistemic vocabulary. The theory I develop assigns more traditional semantic val-ues to logical operators elsewhere. The careful reader will observe that accordingto this theory, logical operators embedding epistemic vocabulary act essentially likeepistemic vocabulary. This observation is implausible unless restricted to logical op-erators embedding epistemic vocabulary, so it is important to bear in mind that moretraditional semantic values for logical operators will be revived in §3.

2.2. A semantics for epistemic possibility and necessity modals

Here are shorthand semantic entries for epistemic possibility and necessity modals:

‘might S’ means that someone accepts that S.

‘must S’ means that everyone accepts that S.

In more formal vocabulary:

[[mighti]]c = [λS . {m : ∃p ∈ gc(i) such that m|p ∈ S}]

[[musti]]c = [λS . {m : ∀p ∈ gc(i), m|p ∈ S}]

Having expanded our lexicon, we can outline a semantics for some nested epistemicmodals. For example, (62) and (63) each mean that everyone accepts that someoneaccepts that we will hire Bob:

(62) It is definitely the case that Bob might be the best candidate for the job.

(63) It must be the case that Bob might be the best candidate for the job.

This shorthand calls attention to an important semantic feature: higher and lower

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epistemic modals need not be associated with the same domain of quantification. Bothlogical operators and modals have indices. Assignment functions map expressionswith different indices to potentially different values. Hence unless expressions areco-indexed, context may contribute different partitions to their interpretation. Forexample, an utterance of (62) may contain modals that are not co-indexed:

(64) It is definitely1 the case that Bob might2 be the best candidate for the job.

The semantic value of (64) is as follows:

[[(64)]]c = {m : ∀p ∈ gc(1), m|p ∈ {m′ : ∃q ∈ gc(2) such that m′|q ∈ [[(65)]]c}},

where (65) is the prejacent of the inner modal in (64):

(65) Bob is the best candidate for the job.

For instance, in a context where (64) is uttered, it could be that the partition gc(1)contains propositions about what sorts of virtues matter when evaluating candidates,while the partition gc(2) contains propositions about which candidates have whatsorts of virtues. In that sort of context, your credences would satisfy (64) just in caseconditional on any proposition about what virtues matter, your credences satisfy thefollowing condition: conditional on some proposition about which candidates havewhich virtues, Bob is the best candidate for the job.

For those especially attentive to syntactic representation: strictly speaking, oursemantics could identify indexed variables as arguments of modals and logical op-erators, rather than indexing these expressions directly. For example, our formalsemantic entry for ‘must’ could be as follows, where v ranges over partitions:

[[must]]c = [λv . λS . {m : ∀p ∈ v, m|p ∈ S}]

In that case, (62) would contain two covert pronouns:

(66) It is definitely v1 the case that Bob might v2 be the best candidate for thejob.

Here the pronouns v1 and v2 denote partitions relative to contexts, according to thefamiliar semantics for referential pronouns, i.e. [[vi]]c = gc(i). The resulting semanticvalue of ‘must vi’ matches the semantic value of ‘musti’ given above. The reader mayreplace expressions of the latter sort with their kosher substitutes throughout.18

18. For simplicity, I will sometimes talk as if the contextually supplied partition is the value of a covert pro-noun. But strictly speaking, I am neutral about the best syntactic implementation of my theory. Partee1989 and Condoravdi & Gawron 1996 have given reasons to doubt that similar implicit arguments are

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2.3. A small detour: advantages of constraining conditional credences

Recall from §1.1 that our use of nested epistemic modals fits naturally with the ideathat sentences constrain imprecise credal states. This idea should seem even morecompelling given all the shorthand just introduced. Suppose we model your mentalstate with a set of probability measures. In other words, suppose we model youas if you have an imaginary mental committee of subjects with precise credences.Then following Rothschild 2012, we could say that sentences constrain your mentalcommittee members, rather than your conditional credences. If a sentence demandsthat everyone accepts a content, for instance, that could just amount to demandingthat each committee member accept that content. In other words, my shorthandsemantic entries for ‘might’ and ‘must’ seem like apt translations of the followingalternative formal semantic entries:

[[might]] = [λS . {I : ∃m ∈ I such that m ∈ S}]

[[must]] = [λS . {I : ∀m ∈ I, m ∈ S}]

Here the variable m ranges over precise credal states, i.e. probability measures, whileS and I range over imprecise credal states, i.e. sets of probability measures. Thisproposal may appear to satisfy many desiderata given in §1. It is worthwhile toreflect on how my semantics differs from this proposal, and especially to notice thatthe imprecise credence proposal is deficient in two respects.

First, on the imprecise credence semantics stated above, embedding a sentenceunder ‘might’ or ‘must’ raises its semantic type. Each modal accepts sets of measuresas inputs and delivers sets of imprecise credal states as outputs. That means thata sentence with a wide-scope ‘might’ or ‘must’ has the wrong semantic type to beembedded under another epistemic modal—a bad result, given our pervasive use ofembedded modals. The most natural repair strategy requires that we model subjectsas having not just imprecise credences, but more complicated mental states. In fact,very complicated mental states are required, since subjects commonly embed epis-temic vocabulary under embedded epistemic vocabulary. For instance, recall that wehave no trouble understanding (49):

(49) If it is probably the case that Jill will live if she jumps, then there is a net.

And deeper embeddings seem perfectly intelligible, as long as the context is richenough to supply the interpretations of relevant expressions. For instance, (49) soundsfine when you are trying to figure out whether there is a net along the roof of your

best analyzed as the values of covert pronouns, and I will not evaluate their arguments in this paper.

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office building. Suppose that the local fire department occasionally puts a trampolineinstead of a net along the roof. Then we are not really licensed in saying that thereis a net along the roof, given just that it is probably the case that Jill will live if shejumps. Instead, we should say something more hedged:

(67) Probably, if it is probably the case that Jill will live if she jumps, then there isa net. (But it might be that there is a trampoline.)

In light of (49) and (67), it is hard to imagine a reason for ruling that embeddingsof epistemic vocabulary beyond a certain level of complexity are are semanticallyuninterpretable. In the absence of such a reason, our theory should deliver semanticvalues for embeddings of arbitrary complexity. Hence in order to repair the imprecisecredence proposal, we would have to model subjects as having not just sets of sets ofmeasures as mental states, but sets of sets of sets of measures, and so on. It is difficultto independently motivate such an arcane model of our mental life.

Second, semanticists like Rothschild must endorse even more complicated mod-els of mental states in order to give a semantics for graded modal vocabulary. It iseasy to imagine existential or universal quantification over members of an imaginarymental committee. But graded modals call for probability measures over committeemembers, and it is difficult to see how one could make sense of this added structurewithin the imprecise credence model without essentially describing subjects as havingprecise credences.

The semantics I defend offers a viable alternative in the neighborhood of theimprecise credence proposal. For starters, the semantics extends naturally to gradedmodals, without requiring that we represent subjects as having mental states morearcane than ordinary credences. As a result, even though it is fairly revisionary to saythat contents of sentences are sets of measures instead of sets of worlds, our model ofcontents can still be defended on the grounds that it simply reflects an independentlymotivated model of our mental life. In addition, according to our semantics, ‘might’,‘must’, and ‘probably’ are all type 〈S, S〉, and ‘if’ is type 〈S, 〈S, S〉〉. Hence complicatedsentences like (67) have well-defined semantic values.

Furthermore, our theory even has the resources to say why complicated sentenceslike (67) might nevertheless sound bad when uttered out of the blue. The same goesfor many sentences containing several referential pronouns. For instance, when ut-tered out of the blue, (68) sounds questionable at best:

(68) ?That made that do that to that.

In particular, sentences with several referential pronouns sound bad in isolation whenthere is a presumption that context will determine different denotations for different

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pronouns. For instance, (68) sounds worse than (69), just as the nested epistemicvocabulary in (70) sounds worse than the repeated unembedded vocabulary in (71):

(68) ?That made that do that to that.

(69) It entered; it saw me; it squealed; and it fainted.

(70) ?Probably, it is probable that probably Jill will probably live.

(71) Jill will probably live; John will probably die; Janet will probably cry; andJoe will probably celebrate.

Context often determines different denotations for pronouns in sentences with nestedepistemic modals. As a result, a rich context is required for the simultaneous inter-pretation of the covert pronouns in sentences such as (67) and (70). Here again, incontrast with semantic injunctions against complicated embeddings, pragmatic ac-counts better fit the contours of our judgments about epistemic vocabulary.

2.4. A semantics for ‘probably’, ‘if’, and a covert type-shifting operator

The expression ‘probably’ has a more complicated semantic function than possibil-ity and necessity modals. The latter modals constrain your credences conditional onpropositions in a contextually determined partition. But as a graded modal, ‘proba-bly’ constrains your credences in members of the partition itself:

[[probablyi]]c = [λS . {m : m(⋃{p ∈ gc(i) : m|p ∈ S}) > .5}]

In our shorthand: find the union of everyone that accepts that S. If you give thatproposition greater than .5 credence, then your credences are contained in the contentof ‘probably S’.19 For example, recall that if we are talking about whether Jill issuicidal, you can correctly assert:

(50) It is probably the case that Jill will live if she jumps.

The partition relevant to the interpretation of ‘probably’ in (50) contains two proposi-tions: either Jill is suicidal or she isn’t. Just one of these propositions accepts that Jillwill live if she jumps, namely the proposition that Jill isn’t suicidal.20 Since you give

19. This semantics follows Kratzer 1991 in taking ‘probably’ to indicate that something is more likely thannot. It is straightforward to adjust the definition so that ‘probably’ instead indicates likelihood above acontextually defined threshold. In a similar vein, it is straightforward to extend the lexicon of this paperto include other simple epistemic vocabulary, such as ‘unlikely’, ‘at least .3 likely’, ‘more likely than’,and comparative epistemic adjectives.

20. A reminder about our shorthand: your credences satisfy the constraint that a proposition accepts acontent just in case your credences conditional on that proposition are contained in that content.

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more than .5 credence to that proposition, your credences are contained in the contentof (50), and that is roughly why it sounds okay for you to say it.

At this point, we can also give a more complete explanation of why the contentof (40) contains your credences about the outcome of rolling a fair die:

(40) It is probably2 even or1 less than four.

As mentioned earlier, the sort of context that is hospitable for (40) makes a certainpartition salient: either the number rolled is low, or it is high. A second partition isalso salient, namely the six possible outcomes of the rolling the die. The first partitiondetermines the content of ‘or’ and the second determines the content of ‘probably’.If you conditionalize your credences on the proposition that the number rolled islow, then you accept that the number is less than four. If you conditionalize yourcredences on the proposition that the number rolled is high, then you have equalcredence in each of the three high number outcomes. Hence you give more than .5conditional credence to the union of outcomes that accept the number rolled is even.That means your credences conditional on the number being high accept that thenumber is probably even. It follows from our semantics for ‘or’ that your credencesare in the content of (40), and that is roughly why it sounds okay for you to say it.

Indicative conditionals are semantically like graded modals, insofar as they alsoconstrain your credences in propositions in contextually determined partitions:

[[ifi]]c = [λS . λT . {m : m(⋃{p ∈ gc(i) : m|p ∈ T}|⋃{p ∈ gc(i) : m|p ∈ S}) = 1}]

In other words, using our shorthand: find the union of everyone that accepts theantecedent of the conditional, and find the union of everyone that accepts the conse-quent. If you have full credence in the latter proposition conditional on the former,then your credences are contained in the content of the conditional itself.21

For example, consider the indicative conditional:

(72) If1 it is high, it is probably2 even.

The context of (72) makes a certain partition salient: either the number rolled is low, orit is high. The former proposition rejects the antecedent of the conditional, while thelatter accepts it. The former proposition also rejects the consequent of the conditional,while the latter accepts it. Hence you have full credence in the union of propositionsthat accept the consequent of (72), conditional on the union of propositions that accept

21. A disclaimer: this semantics is sufficient to address the motivating concerns of the present paper, but itis not my final word on indicative conditionals. I defend an alternative probabilistic semantics in Moss2014, motivated by concerns that I have bracketed for ease of exposition here.

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the antecedent. It follows from our semantics for ‘if’ that your credences are in thecontent of (72), and that is roughly why it sounds okay for you to say it.

There is one important respect in which our theory so far is incomplete. I havenot yet given a semantics for simple sentences such as:

(65) Bob is a hire.

(73) Jill jumps.

(74) The number rolled is high.

For instance, I have said certain partition propositions “accept that the number rolledis high” or “accept the antecedent of ‘if it is high, it is probably even’.” This isshorthand for a constraint on probability measures, namely that after conditionalizingon the partition proposition, the resulting measure is contained in the content of (74).Hence simple sentences like (74) must have sets of measures as their contents.

There is a natural way of associating simple sentences with sets of measures.According to standard truth conditional semantic theories, the content of a simplesentence is a set of worlds. According to my theory, the content of a simple sentenceis the set of measures that assign probability 1 to that set of worlds.22 This meansthat the theory need not start from scratch to deliver semantic values for referringexpressions, predicates, quantifiers, and so forth. Instead, a covert operator convertstraditional semantic values into alternative semantic values:

[[C]]c = [λp . {m : m(p) = 1}]

For example, the logical form of the sentence ‘Jill jumps’ is more accurately repre-sented as ‘C Jill jumps’. The semantic value of this sentence is a set of measures,namely {m : m({w . Jill jumps in w}) = 1}. Since simple sentences accompanied bythe covert operator C have sets of measures as semantic values, simple sentences canbe arguments of type 〈S, S〉 operators and type 〈S, 〈S, S〉〉 operators.

Furthermore, arguments of logical operators can include both simple sentencesand sentences containing epistemic vocabulary. For example, the logical form of (40)is more accurately represented as follows:

(40) [ probably2 [ C [ it is even ] ] ] or1 C [ it is less than four ].

This detail lets us finally give a complete explanation of why your credences arecontained in the content of (40) in the context described above. In our most recent

22. This content may seem inappropriate, since giving full credence to some proposition is a very strongconstraint. In short, I have made some assumptions in order to simplify my discussion, and refinementsof the theory in §3 address this worry. For a more thorough treatment of these issues, see chapter 2 ofMoss 2014.

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explanation of this fact, we said that “if you conditionalize your credences on theproposition that the number rolled is low, then you accept that the number is lessthan four.” The more complete explanation replaces this with the following claim: ifyou conditionalize your credences on the proposition that the number rolled is low,then the resulting credence distribution has full credence that the number is less than four.Fans of gory detail should see Appendix B.1 for an explanation in formal vocabulary.

To sum up so far: I have introduced a semantics for eight expressions, includingbasic logical operators and epistemic vocabulary. According to this theory, there is asense in which logical operators are epistemic vocabulary. If they occur in the midstof epistemic modals, logical operators deliver constraints on credences that dependon what is accepted by propositions in contextually determined partitions. Assigningthe same sort of semantic values to logical operators and epistemic vocabulary helpsexplain the behavior of the latter. The way that ‘might’ and ‘must’ and ‘probably’interact with each other has a lot in common with the way they interact with logicaloperators. According to my theory, this is to be expected, as both are interactionsbetween different sorts of epistemic vocabulary.

3. A number of refinements and explanations

I have made three simplifying assumptions in developing the semantics in §2. Inorder to refine the semantics, I will identify these assumptions and say how they canbe removed. The first is about the standard effect of assertion, namely that whenyou hear an assertion with a certain content, you generally come to have credencescontained in that content. This claim abstracts away from lying, pretense, supposition,and so on. But more importantly, even in normal cases of assertion, your credencesdo not really come to be contained in asserted contents. The contents of sentencesare simply too strong to play that role. The content of a simple sentence is the set ofmeasures that assign probability 1 to some proposition. But it is arguably almost neverrational to have full credence in a proposition. Having full credence in a propositionmakes you bet on that proposition at arbitrarily risky odds, and makes your beliefin that proposition rationally unrevisable by conditionalizing on further evidence. Inother words, it makes you have blind faith in a proposition. Assertions rarely if everhave such a dramatic effect.

It might be possible to answer this complaint by saying that our theory governsideal cases, and that ideal communication really does make subjects have full cre-dence in asserted contents. But even this answer should be accompanied by somesuggestions about the effect of assertion in realistic cases. Here is one suggestion: as

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far as the conversational record is concerned, an act of assertion is a proposal thatthe content of the assertion be accepted for conversational purposes. For example,suppose you assert that it is raining. Then it will sound bad for either of us to say oreven suppose that it might not be raining:

(75) a. Alice: Oh no. It is raining.b. Bob: #If it might not be raining, we should buy some sunglasses.

If your assertion is not challenged or retracted, it does seem that we accept its strongcontent for conversational purposes. Having accepted that content, Alice and Bob doresemble subjects who would accept bets at arbitrary odds, conversationally speaking,as they cannot even raise the possibility that it is not raining.23

In addition to affecting the conversational record, an assertion affects conversa-tional participants. An assertion does not exactly affect your credences, but somethingmore like your credences for practical purposes. For example, an assertion of (75-a) mayhave the effect that for practical purposes, it is just as if your credences are containedin its content—i.e. when it comes to your preferences and decisions, it is just as ifyou have full credence in the proposition that it is raining. This account of assertionis designed to mimic contemporary accounts of full belief according to which youbelieve a proposition when you can treat it as certain for practical purposes. For in-stance, according to Weatherson 2005, you believe a proposition roughly just in caseconditionalizing on that proposition changes none of your preferences over salientoptions.24 The analogous account of assertion says you accept an assertion just incase updating your credences on its content changes none of your preferences oversalient options. For example, you accept (75-a) just in case updating on the proposi-tion that it is raining changes none of your preferences over salient options. In otherwords, given the analogous account of full belief, you accept (75-a) just in case youbelieve that it is raining. This seems like exactly the right result, as assertions of sim-ple sentences are traditionally taken to constrain your full beliefs. To sum up: giventhe above accounts of full belief and assertion, you accept an assertion of a simplesentence just in case you believe its content. Even if our accounts of full belief andassertion must ultimately be modified, the latter will deliver intuitive results as longas it mirrors the former.

The second simplifying assumption made in §2 is that logical operators have justone semantic value each. In fact, my theory requires a serious and significant revision

23. This effect of assertion on the conversational record is elegantly explained by models on which thecontext set itself is fine-grained. For further discussion, see the context probabilism introduced in §8 ofYalcin 2007.

24. Cousins of this principle are defended by Williamson 2000, Ganson 2008, Fantl & McGrath 2010,and Schroeder & Ross 2014.

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of this assumption, namely that logical operators have different types of semanticvalues, depending on whether they embed non-epistemic or epistemic vocabulary.For example, the semantic value of negation given in §2 must have a different type ofsemantic value of negation in simple sentences such as:

(76) John does not smoke.

For suppose (76) has the following logical form:

(77) not1 [ C John smokes ]

Then according to the semantics for ‘not’ in §2, the content of (76) contains yourcredences just in case there is no proposition in the relevant partition such that youhave full credence that John smokes, given that proposition. This is not at all what (76)intuitively means. For many partitions, it is very easy for your credences to satisfythis constraint, even if you have a relatively high credence that John smokes. It shouldintuitively be much harder for your credences to be contained in the content of (76).In fact, in light of our semantics for other sentences without epistemic vocabulary, thecontent of (76) should intuitively be the set of measures that assign probability 1 tothe proposition that John does not smoke.

The appropriate refinement of our semantics involves distinguishing between log-ical operators that embed epistemic vocabulary and logical operators that embed sim-ple sentences. A simple sentence actually has a set of worlds as its semantic value,which can serve as the argument of a covert type-raising operator. This covert oper-ator need not occur immediately above every simple sentence. In our refined seman-tics, logical operators can have sets of worlds as arguments. In addition to reinstatingtraditional semantic values for simple sentences, we reinstate traditional semantic val-ues for logical operators, adding these values to those introduced in §2. Hence logicaloperators have different semantic values in different linguistic contexts: traditionalvalues when their arguments are sets of worlds, and our §2 semantic values whentheir arguments are sets of measures. The logical form of ‘John does not smoke’ is(78) rather than (76):

(76) not1 [ C John smokes ]

(78) C [ not John smokes ]

The sentence under the covert operator has a set of worlds as its semantic value,namely the proposition that John does not smoke. Hence the content of (76) is the setof measures that assign probability 1 to that proposition, as desired.

This refinement of our semantics addresses several other potential problems as

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well. For instance, suppose the logical form of (79) is given by (80):

(79) John smokes or Jill drinks.

(80) [ C John smokes ] or1 [ C Jill drinks ]

Then if the content of (79) contains your credences, there must be some contextuallydetermined partition such that conditional on each proposition in the partition, youeither have full credence that John smokes or full credence that Jill drinks. But in-tuitively you can utter a disjunction even if no such propositions would make yousure of either disjunct. In addition, our semantics should predict that the followinginference is valid:

(81) a. It is not the case that John does not smoke.b. Hence: John smokes.

And likewise for the following:

(82) a. It is not the case that both John smokes and Jill drinks.b. Hence: either John does not smoke or Jill does not drink.

However, from the premise that no one accepts that no one accepts that John smokes,we cannot generally infer that John smokes. From the premise that no one acceptsthat everyone accepts both that John smokes and Jill drinks, we cannot generally inferthat everyone either accepts: (a) that no one accepts that John smokes, or (b) thatno one accepts that Jill drinks. In other words, if the covert type-raising operator ‘C’occurs immediately above ‘John smokes’ and ‘Jill drinks’ in (81) and (82), the result-ing inferences are invalid. Hence our §2 semantics does not automatically validatedouble negation elimination or applications of De Morgan’s Laws, even restricted toinferences not containing any epistemic vocabulary.

The above refinement of our semantics validates instances of these inferenceswhere appropriate. For instance, the logical form of ‘John smokes or Jill drinks’ isgiven by (83):

(83) C [ John smokes or Jill drinks ]

The semantic value of (83) is the set of measures that assign probability 1 to theproposition that either John smokes or Jill drinks. This semantic value may containyour credences even if no salient information would make you sure of either disjunct.The logical form of the double negation elimination argument is not (84) but (85):

(84) a. not1 not2 C John smokesb. Hence: C John smokes

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(85) a. C not not John smokesb. Hence: C John smokes

The logical form of the De Morgan’s Law argument is not (86) but (87):

(86) a. not1 [ C John smokes and2 C Jill drinks ]b. Hence: [ not3 C John smokes ] or4 [ not5 C Jill drinks ]

(87) a. C not [ John smokes and Jill drinks ]b. Hence: C [ [ not John smokes ] or [ not Jill drinks ] ]

It is not hard to verify that the latter inferences are valid, as desired.I should emphasize that on the semantics developed here, logical operators are

polymorphic. This claim constitutes a loss of theoretical parsimony, which some read-ers may count as a cost of my theory. Some may even judge that this cost is ultimatelytoo great to be outweighed by the benefits of the theory. However, several facts mayhelp mitigate this cost. For starters, it is a familiar observation that logical opera-tors can embed expressions of various semantic types; indeed, “virtually every majorcategory can be conjoined with ‘and’ and ‘or’” (Partee &